Annals of Probability

A Gaussian kinematic formula

Jonathan E. Taylor

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In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form fp=F(y1(p),…,yk(p)) for FC2(ℝk;ℝ) and (y1,…,yk) a vector of C2 i.i.d. centered, unit-variance Gaussian fields.

The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest.

As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets Mf−1[u,+∞)=My−1(F−1[u,+∞)) of the field f.

The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.

Article information

Ann. Probab., Volume 34, Number 1 (2006), 122-158.

First available in Project Euclid: 17 February 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields 53A17: Kinematics 58A05: Differentiable manifolds, foundations
Secondary: 60G17: Sample path properties 62M40: Random fields; image analysis 60G70: Extreme value theory; extremal processes

Random fields Gaussian processes manifolds Euler characteristic excursions Riemannian geometry


Taylor, Jonathan E. A Gaussian kinematic formula. Ann. Probab. 34 (2006), no. 1, 122--158. doi:10.1214/009117905000000594.

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  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester.
  • Adler, R. J. (2000). On excursion sets, tube formulae, and maxima of random fields. Ann. Appl. Probab. 10 1–74.
  • Adler, R. J. and Taylor, J. E. (2006). Random Fields and Geometry. Birkhäuser, Boston. To appear. Preliminary versions of Chapters 1–12 available at
  • Airault, H. (1991). Differential calculus on finite codimensional submanifolds of the Wiener space–-the divergence operator. J. Funct. Anal. 100 291–316.
  • Airault, H. and Malliavin, P. (1988). Intégration géométrique sur l'espace de Wiener. Bull. Sci. Math. (2) 112 3–52.
  • Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
  • Bernig, A. and Br öcker, L. (2002). Lipschitz–Killing invariants. Math. Nachr. 245 5–25.
  • Cao, J. and Worsley, K. J. (1999). The detection of local shape changes via the geometry of Hotelling's ${T}^2$ fields. Ann. Statist. 27 925–942.
  • Chern, S.-S. (1944). A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds. Ann. of Math. (2) 45 747–752.
  • Chern, S.-S. (1966). On the kinematic formula in integral geometry. J. Math. Mech. 16 101–118.
  • Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418–491.
  • Gray, A. (1990). Tubes. Addison–Wesley Publishing Company Advanced Book Program, Redwood City, CA.
  • Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge Univ. Press.
  • Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge Univ. Press.
  • Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23 608–639.
  • Takemura, A. and Kuriki, S. (2002). Maximum of Gaussian field on piecewise smooth domain: Equivalence of tube method and Euler characteristic method. Ann. Appl. Probab. 12 768–796.
  • Taylor, J., Takemura, A. and Adler, R. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362–1396.
  • Taylor, J. E. and Adler, R. J. (2003). Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 533–563.
  • Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61 461–472.
  • Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of $\chi^2$, $F$ and $t$ fields. Adv. in Appl. Probab. 26 13–42.
  • Worsley, K. J. (1995). Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. in Appl. Probab. 27 943–959.
  • Worsley, K. J. (1995). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist. 23 640–669.
  • Worsley, K. J. (1996). The geometry of random images. Chance 9 27–40.
  • Worsley, K. J. and Friston, K. J. (2000). A test for a conjunction. Statist. Probab. Lett. 47 135–140.